Although there is a theory of non-commutative polynomial rings, it presents some difficulties and will not be treated on this page. Thus, we will work only with commutative rings for their polynomial rings.

The degree of a polynomial ${\displaystyle a_{0}+a_{1}X+...+a_{n}X^{n}}$ is defined to be ${\displaystyle n}$. If ${\displaystyle R}$ is a field, and ${\displaystyle f}$ and ${\displaystyle g}$ are polynomials of ${\displaystyle R[X]}$, then we can divide ${\displaystyle f}$ by ${\displaystyle g}$ to get ${\displaystyle f=gq+r}$. However, we can also do this for any arbitrary ring if the leading coefficient of ${\displaystyle g}$ is 1.